Yes to and from the reception. There will be 2 pick up points, one at The Surf Club (downtown Montauk) and one at Montauk Manor.
Yes, there is uber in Montauk as well as local taxis. 668 MTK Taxi: (631) 668-1040 Moko Taxi: (631) 668-7070 Surf Taxi: 631-668-2777 & more!
Driftwood Resort, A Wave Inn, and Beach Plum resort currently offer 1-night stays - we recommend booking through Expedia. There are also a number of hotels in other towns in the Hamptons that are a short drive away (The Ocean Colony in Amagansett, East Hampton House Resort, The Huntting Inn in EH, The Maidstone in EH, East Hampton Art House).